### A bound for the average rank of a family of abelian varieties

In this note, we consider a one-parameter family of Abelian varieties $A/\mathbb{Q}\left(T\right)$, and find an upper bound for the average rank in terms of the generic rank. This bound is based on Michel's estimates for the average rank in a one-parameter family of Abelian varieties, and extends previous work of Silverman for elliptic surfaces.